How to non-destructively bevel and extrude a mesh intersection?
If I understand your problem correctly, you should be able to solve this by using the bevel modifier limited to a vertex group.
First, go to the Object Data tab in the properties pane (the green one, can't miss it) and open Vertex Groups.  Hit the &quot;+&quot; button to create one.

Then, select the vertices you want to bevel in Edit mode.  While the group you just created is selected, click the &quot;Assign&quot; button underneath the vertex group list.

Third, go to the Modifiers tab (the one with the wrench) and add a Bevel modifier.  In the &quot;Limit Method&quot; drop-down, select Vertex Group.

Finally, select the vertex group you created earlier.  Play with the numbers a bit, and you should be good!